How do you make them mandatory without privacy invasion? You must keep records on who is unvaccinated.
The precursor to every 20th century atrocity was diving the population by decree.
The QR codes themselves enable businesses and government to track your detailed movements every day though. This is why that's much worse.
For example, what if guests could regenerate the QR code, or the app did it automatically, every five minutes? And the QR code contained nothing but a cryptographically verifiable assertion that the person presenting it is vaccinated, together with a picture (or a reference to a downloadable picture) of the guest? That would prevent the host from being able to gather anything beyond what they already can from their CCTV, while getting the vaccination assurance they need.
Trouble is, I can't think of a way to do this without connectivity, and that's problematic due to downtime, service availability, and so on. A privacy eroding fallback to an offline system would be possible, but runs the risk of privacy violators simply "accidentally" losing connectivity.
Anyway, there's a challenge for HN. Design such a system :-)
Of course that also presents issues with people not owning smartphones, not knowing how to use them or having dead batteries as you mention already.
But what you describe would be better than the current setup if the operator is really transparent about how any personal info is used, yes.
All the covid tracking apps during the first wave were really privacy conscious (avoiding personal info, using open source etc) but the whole QR thing as introduced in Europe doesn't seem to have taken this into account at all. Of course that's because it's more personal in general but the whole open source thing has been dropped too. Only Switzerland seems to be doing that now.